#199765

order	512
nilpotency class	3
isoclinism	isoclinic class of #199546
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,10,7,12,5,11)(3,13,17,23,15,26,16,8)(4,18,20,9,21,24,19,25)(14,22,30,32,28,31,29,27)(33,34)(35,39,45,54)(36,46,48,40)(37,49)(38,42)(41,51)(43,52)(44,56,50,53)(47,55,62,60)(57,64)(58,59)(61,63);(1,3)(2,8,12,23)(4,14)(5,16)(6,17)(7,15)(9,22,25,31)(10,13,11,26)(18,27,24,32)(19,29)(20,30)(21,28)(33,35)(34,39,42,54)(36,43)(37,50)(38,45)(40,52,46,59)(41,56,49,53)(44,51)(47,61)(48,58)(55,64,60,63)(57,62);(1,4,7,21)(2,9,12,25)(3,14,15,28)(5,20,6,19)(8,22,23,31)(10,18,11,24)(13,27,26,32)(16,30,17,29)(33,36,38,48)(34,40,42,46)(35,43,45,58)(37,47,51,62)(39,52,54,59)(41,55,49,60)(44,57,50,61)(53,63,56,64);(3,15)(8,23)(13,26)(14,28)(16,17)(22,31)(27,32)(29,30)(33,37,38,51)(34,41,42,49)(35,44,45,50)(36,47,48,62)(39,53,54,56)(40,55,46,60)(43,57,58,61)(52,63,59,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106961);